Problem: $f(x,y) = x^3 - 2x^2 + x$ What is $\dfrac{\partial f}{\partial y}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $3x^2 - 4x$ (Choice B) B $0$ (Choice C) C $x^2 - 2x + 1$ (Choice D) D $3x^2 - 4x + 1$
Solution: We want to find $\dfrac{\partial f}{\partial y}$, which is the partial derivative of $f$ with respect to $y$. When we take a partial derivative with respect to $y$, we treat $x$ as if it were a constant. Let's break $f(x, y)$ down term by term. $\begin{aligned} &\dfrac{\partial}{\partial y} \left[ x^3 \right] = 0 \\ \\ &\dfrac{\partial}{\partial y} \left[ -2x^2 \right] = 0 \\ \\ &\dfrac{\partial}{\partial y} \left[ x \right] = 0 \end{aligned}$ Adding the terms back together, we get the partial derivative. In conclusion: $\dfrac{\partial f}{\partial y} = 0 + 0 + 0 = 0$